Two Dimensional Rod Resonator for RF Filtering

ABSTRACT

A microelectromechanical resonator device is provided having two-dimensional resonant rods. The resonator device has a piezoelectric layer formed with a plurality of alternating rods and trenches. A bottom electrode is in contact with a bottom surface of the piezoelectric layer. A top electrode metal grating of conductive strips is aligned in contact with corresponding rods of the piezoelectric layer.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority under 35 U.S.C. § 119(e) of U.S.Provisional Application No. 62/935,769, filed on 15 Nov. 2019, entitled“Two Dimensional Rod Resonator for RF Filtering,” and of U.S.Provisional Application No. 62/984,426, filed on 3 Mar. 2020, entitled“Two Dimensional Rod Resonator for RF Filtering,” the disclosures ofwhich are hereby incorporated by reference.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

N/A

BACKGROUND

In the last decades, acoustic resonators and filters based on aluminumnitride (AlN) film-bulk-acoustic-resonators (FBARs) have been largelyused to form filters in commercial RF front-ends. Such devices excite athickness-extensional (TE) mode of vibration. So, despite theirhigh-k_(t) ² (˜7%), the resonance frequency (f_(res)) of FBARs is set bythe thickness of their AlN plate, thus being lithographically tunableonly through a large fabrication complexity. Recently,two-dimensional-mode-resonators (2DMRs) were discovered. 2DMRs excite aset of Lamb-waves, in suspended AlN plates. To do so, they rely on twoidentical metallic gratings connected to opposite voltage polarities.The use of these gratings is key to constrain the excited modes in themetallized AlN regions. However, since these modes are significantlycoupled along the lateral direction, 2DMRs can only attain a k_(t)²-value approaching 5%, thus not being suitable to form filters forwideband applications.

SUMMARY

A class of aluminum nitride resonators, labeled two-dimensional resonantrods (2DRRs), for operation in the radio-frequency (RF) range, isdescribed, and a prototype is demonstrated. 2DRRs rely on the excitationof an array of locally resonant rods, micro-fabricated in thin-film AlNplates. With this technology, a low-impedance ((ωC0)⁻¹≈208 Ω) 2DRR hasbeen built, operating at 2.35GHz and showing an electromechanicalcoupling coefficient (k_(t) ²), for lithographically defined AlNresonators, exceeding 7.4%. Such a high k_(t) ²-value, that matchesclosely the predicted value through finite element methods (FEM), can beattained while relying on an un-patterned bottom metal plate. Thisfeature permits to simultaneously attain large capacitance values and anoptimal AlN crystalline orientation. Moreover, as verified through FEM,the resonance frequency of 2DRRs can be lithographically tuned without asubstantial degradation of k_(t) ².

Further aspects include the following:

1. A resonator device comprising:

-   -   a piezoelectric layer suspended from a substrate, the        piezoelectric layer having a width direction, a length        direction, and a thickness direction, the piezoelectric layer        having a bottom surface and a top surface, the piezoelectric        layer comprising a beam extending along the bottom surface        continuously in the width direction and the length direction, a        plurality of rod portions extending continuously in the length        direction and upwardly from the beam to the top surface, the rod        portions spaced apart in the width direction;    -   a bottom electrode comprising a metal plate in contact with the        bottom surface of the piezoelectric layer; and    -   a top electrode comprising a metal grating comprising a        plurality of conductive strips on the top surface of the        piezoelectric layer, each of the conductive strips aligned in        contact with a corresponding one of the rod portions of the        piezoelectric layer;    -   wherein the plurality of conductive strips and corresponding        ones of the rod portions of the piezoelectric layer form a        plurality of rods, and portions of the beam of the piezoelectric        layer and the bottom electrode form a plurality of trenches        between the rods.

2. The resonator device of 1, wherein a thickness of each of the rods isgreater than a thickness of each of the trenches.

3. The resonator device of any of 1-2, wherein a thickness of each ofthe rod portions of the piezoelectric layer is greater than a thicknessof the piezoelectric layer of each of the trenches.

4. The resonator device of any of 1-3, wherein each of the trenches hasa depth in the thickness direction selected to constrain acoustic energywithin the rods.

5. The resonator device of any of 1-4, wherein each of the rods has awidth selected to provide a determined frequency.

6. The resonator device of any of 1-5, wherein each of the trenches hasa width selected to optimize the determined frequency of the rods.

7. The resonator device of any of 1-6, wherein the rods and the trencheseach have an equal width within ±1%.

8. The device of any of 1-7, further comprising circuitry incommunication with the resonator device to apply an alternating voltagethrough the top electrode and the bottom electrode to excite one or moredilatational modes of mechanical vibration in the rods of thepiezoelectric layer.

9. The resonator device of any of 1-8, wherein the piezoelectric layeris selected from aluminum nitride, lead zirconate titanate, lithiumniobate, lithium tantalite, zinc oxide, gallium nitride, scandiumnitride, aluminum scandium nitride, and quartz, and combinationsthereof.

10. The resonator device of any of 1-9, wherein the top electrode andthe bottom electrode are each selected from aluminum, platinum,ruthenium, molybdenum, tungsten, and gold, and combinations thereof.

11. The resonator device of any of 1-10, wherein the piezoelectric layeris aluminum nitride, the bottom electrode is platinum, and the topelectrode is aluminum.

12. The resonator device of any of 1-11, wherein the device has anelectromechanical coupling coefficient between electrical and acousticenergy of at least 7%.

13. A filter comprising the resonator device of any of 1-12, andelectrical connections to the resonator device.

14. A radio frequency (RF) component including the resonator device ofany of 1-13, and electrical connections to the resonator device, whereinthe RF component is a filter, oscillator, synthesizer, sensor, coupler,or transformer.

15. A method of fabricating the resonator device of any of 1-14,comprising:

-   -   forming a bottom metal layer on a substrate to form the bottom        electrode;    -   forming a piezoelectric material on the bottom metal layer;    -   forming a top metal layer on the piezoelectric material        comprising a grating of conductive strips to form the top        electrode;    -   removing material from the piezoelectric material between the        conductive strips of the top electrode to form the plurality of        alternating rods and trenches; and    -   releasing the bottom electrode, the piezoelectric material and        the top electrode from the substrate.

16. A method of fabricating a resonator device, comprising:

-   -   forming a bottom metal layer on a substrate to form a bottom        electrode;    -   forming a piezoelectric material on the bottom metal layer;    -   forming a top metal layer on the piezoelectric material        comprising a grating of conductive strips to form a top        electrode;    -   removing material from the piezoelectric material between the        conductive strips of the top electrode to form a plurality of        alternating rods and trenches; and    -   releasing the bottom electrode, the piezoelectric material and        the top electrode from the substrate.

17. The method of any of 15 and 16, wherein the bottom electrode isformed from aluminum, platinum, ruthenium, molybdenum, tungsten, orgold, or combinations thereof

18. The method of any of 15-17, wherein the top electrode is formed fromaluminum, platinum, ruthenium, molybdenum, tungsten, or gold, orcombinations thereof.

19. The method of any of 15-18, wherein the piezoelectric material isformed from aluminum nitride, lead zirconate titanate, lithium niobate,lithium tantalite, zinc oxide, gallium nitride, scandium nitride,aluminum scandium nitride, or quartz, or combinations thereof.

20. The method of any of 15-19, further comprising forming one or morevias in the piezoelectric layer to the bottom electrode, and depositinga metal material in the one or more vias and on device pads.

DESCRIPTION OF THE DRAWINGS

FIG. 1A is a schematic view of a 2.3 GHz two-dimensional rod resonator(2DRR) according to the technology described herein.

FIG. 1B illustrates a cross-sectional view of a periodic unit-cellforming a device according to the technology herein and a description ofmaterials adopted in one 2DRR implementation (T_(AlN) ⁽¹⁾=400 nm,T_(AlN) ⁽²⁾=600 nm, T_(m) ⁽¹⁾=250 nm and T_(m) ⁽²⁾=330 nm).

FIG. 1C illustrates adopted nomenclature for the three main regionsforming the 2DRR unit-cell and schematic representation of the differentdisplacement components defined in Eqs. (2-3).

FIG. 2 illustrates a schematic view of the fabrication flow used tobuild a 2DRR as described herein. In the example illustrates, theprocess starts with the sputtering deposition of a 250-nm-thick Ptfull-plate used as bottom electrode (a) followed by a 1-μm-thickAlN-film deposition (b). Vias are then formed, in the AlN-layer, througha wet-etch process (c) followed by the sputtering and lift-off of a 330nm-thick Al-layer forming the top metal layer (d). Then, a 150-nm-thickAu-layer is deposited, on the device pads and in the vias (e), so as tominimize their associated ohmic losses. Later, the releasing holes areformed through an AlN dry-etch (f) followed by the patterning of the topelectrodes and the derivation of the AlN trenches (g). These two stepsare attained through a combination of wet and dry-etch, which isoptimized so as to minimize the surface roughness and optimize the AlNsidewall angle in the trenches. Finally, the devices are releasedthrough a XeF₂ silicon etching process (h).

FIG. 3 illustrates scanned Electron Microscope (SEM) pictures of afabricated 2DRR (Device A). The picture on the left shows the devicetop-view. A zoom-in picture of the part highlighted in dashed lines isshown on the right. This picture shows the device topology after theformation of rods and trenches.

FIG. 4 illustrates: a) Simulated trends of φ and Γ_(real) vs. f andrelative to the unit-cell of the fabricated 2DRR; b) analyticallyderived trends of T (in blue) and R (i.e. 1-|T|, in red) vs. f relativeto the fabricated 2DRR (FIG. 1A) and for f varying around the measuredf_(res); c) Analytically derived k_(ef-real) (in blue) and k_(ef-im) (inbrown), when assuming the same geometrical and material characteristicsadopted in the fabricated device.

FIG. 5A illustrates FEM simulated resonant total displacement modaldistribution, relative to the fabricated 2DRR (FIG. 1A).

FIG. 5B illustrates measured (solid line) and simulated (dashed line)admittance for the fabricated device (FIG. 1A). As evident, with theexception of few addition spurious resonances in the measured results,there is a close matching between measured and FEM simulated results.Scanned Electron Microscope (SEM) pictures of the fabricated device arealso shown.

FIG. 5C illustrates FEM simulated distributions of f_(res) and k_(t) ²vs. a, when assuming the same material stack and geometrical parametersused for the fabricated 2DRR device (FIG. 1).

FIG. 6 illustrates analytically simulated distribution of the imaginarypart of Z_(b) as the frequency (f) is varied between 0 and 6 GHz.

FIG. 7 illustrates simulated trend of T vs. f when assuming the samematerial and geometrical parameters used for considered unit-cell (seeFIGS. 1B, 1C) and when considering each edge connected to an infinitesequence of unit cells.

FIG. 8 illustrates FEM simulated modeshape of the total displacement ina chain of 7 unit-cells (FIGS. 1B, 1C) when driven, at 2.4 GHz, by onepiezoelectric generator attached to it (here an electrically driven 2DRRunit-cell was used as the generator at 2.4 GHz). The spatialtotal-displacement distribution along a cut line, starting from thetop-left edge of the closest unit-cell to the generator and ending atthe top-right corner of the furthest PML, is also reported.

FIG. 9 illustrates FEM simulated modeshape of the total displacement ina chain of 7 unit-cells (FIGS. 1B, 1C), when driven, at 1.7 GHz, by onepiezoelectric generator attached to it. The spatial total displacementdistribution along a cut line, starting from the top-left edge of theclosest unit-cell to the generator and ending at the top-right corner ofthe furthest PML, is also reported.

FIG. 10 illustrates FEM simulated modeshape of the total displacement ina chain of 7 unit-cells (FIGS. 1B, 1C), when this is driven, at 0.5 GHz,by one piezoelectric generator attached to it. The spatial totaldisplacement distribution along a cut line, starting from the top-leftedge of the closest unit-cell to the generator and ending at thetop-right corner of the furthest PML, is also reported.

FIG. 11 illustrates measured (solid line) and FEM simulated (dashedline) admittance of a fabricated AlN device having active regioncharacterized by the same material stack used by the fabricated 2DRR,but not using trenches between adjacent top metal strips. This deviceshows kt², resistance at resonance (R_(tot)=R_(m)+R_(s)), loaded qualityfactor Q_(3dB) (extracted from the 3 dB bandwidth) and C₀ in excess of4.4%, 64 Ω, 236 and 330 fF, respectively. The measured k² (4.6%) and C₀(340 fF) values match closely their FEM predicted values. Also, themeasured Q_(3dB)-value matches well the one attained by the reported2DRR. This fact clearly shows that the quality factor found for bothdevices is not limited by the presence of the trenches.

FIG. 12 illustrates a) Schematic views relative to the active region ofthe reported 2DRR (FIG. 13). The device is formed by a sets ofmetallized AlN rods (labeled “rods”) connected through thinner AlN-beams(labeled “trenches”) attained by forming trenches between the rods; b)Cross-sectional view of the active region of the reported 2DRR; c) FEMsimulated dispersion curves of the frequency vs. lateral wavevector(k_(x)) relative to the S1-mode, for rods (line of closed circles) andtrenches (line of open circles). As evident, a large isolation gapexists between the two derived curves. This feature makes the trenchesunable to guide the modal energy created in the rods. So, as a firstorder of approximation, the trenches act as fixed constraints, thustrapping the acoustic energy in the rods. Consequently, the acousticwavelength in the 2DRR becomes set by the width (W_(e)) of the rods.

FIG. 13 illustrates measured (solid line) and FEM simulated (dashedline) admittance response for a fabricated 2DRR (Device A) (FIGS. 2-3).The device shows measured k_(t) ², resistance at resonance(R_(tot)=R_(m)+R_(s)), loaded quality factor Q_(3dB) (extracted from the3 dB bandwidth) and C₀ in excess of 7.4%, 56 Ω, 185 and 325 fF,respectively. The measured k_(t) ² and C₀ match closely their FEMpredicted values (7.2% and 300 fF, respectively). The device simulatedmodeshapes relative to its total, vertical and lateral resonantdisplacement components are also reported.

FIG. 14 illustrates: a) Measured (solid line) and FEM simulated (dashedline) admittance of a fabricated AlN device (Device B) having activeregion characterized by the same material stack used by the fabricated2DRR (FIGS. 2-3; 13), but not using trenches between adjacent top metalstrips. This device, which was fabricated on the same silicon wafer usedfor the reported 2DRR (FIG. 13), shows k_(t) ², resistance at resonance(R_(tot)=R_(m)+R_(s)), loaded quality factor Q_(3dB) (extracted from the3 dB bandwidth) and C₀ in excess of 4.4%, 64 Ω, 236 and 330 fF,respectively. The measured k_(t) ² and C₀ values match closely their FEMpredicted values (4.1%, 340 fF respectively). Also, the measuredQ_(3dB)-value matches well the one attained by the reported 2DRR (FIG.13). This fact clearly shows that the quality factor found for bothdevices is not limited by the presence of the trenches.

FIG. 15: FEM simulated trends of f_(res) and k_(t) ² vs. W_(e) (FIG.12). As evident, 2DRRs simultaneously enable a significant lithographicfrequency tunability (>117 MHz) and a large k_(t) ² exceeding 5%. Thisfeature renders them suitable components to form monolithic integratedwideband filters for next-generation RF front-ends. As an example, thetransmission scattering parameter (S₂₁) of a 5^(th)-order 50 Ω-matchedladder filter using 2DRRs exhibiting the same Q_(3dB)-value found inthis work and a k_(t) ²-value extracted from FEM (see picture on theleft) is reported. The simulated S₂₁-curve (see picture on the right)shows that 2DRRs enable RF filters simultaneously exhibitinginsertion-loss (IL) lower than 2.9 dB, out-of-band rejection exceeding30 dB and a wide fractional bandwidth (>3.7%).

DETAILED DESCRIPTION

A class of AlN resonators is described that can provide acousticresonant components that simultaneously exhibit high k_(t) ² and alithographic frequency tunability. These radio-frequency devices,labelled as two-dimensional-resonant-rods (2DRRs), exploit theunconventional acoustic behavior exhibited by a forest of locallyresonant rods, built in the body of a profiled AlN layer that issandwiched between a bottom un-patterned metal plate and a top metallicgrating. 2DRRs exhibit unexplored modal features that make them able toachieve high-k_(t) ², a significant lithographic frequency tunabilityand a relaxed lithographic resolution, while relying on an optimal AlNcrystalline orientation. The operation of 2DRRs is discussed with use ofanalytical and finite-element (FE) investigations. The measuredperformance of a fabricated 2DRR, showing a k_(t) ² in excess of 7.4%,is also reported.

By way of further explanation, in the last decades, bulk-acoustic-wave(BAW) filters have become essential components of 1G-to-4G radios. Thesedevices rely on the high electromechanical coupling coefficient (k_(t)²˜7%), attained by aluminum nitride (AlN) film-bulk-acoustic-resonators(FBARs), to achieve a wideband and low-loss frequency response. As theresonance frequency of FBARs is set by their thickness, the integrationof multiple FBARs, to form filters, can only be attained through theadoption of frequency tuning fabrication steps, such as mass loading ortrimming. However, as the ability to reliably control these stepssignificantly decays for thinner FBARs, manufacturing FBAR-based filtersthat can address the needs of emerging IoT and 5G high-frequencyapplications is becoming more and more challenging.

More particularly, acoustic resonators and filters have represented keycomponents for several radio-frequency (RF) applications and systems.For instance, their superior performance, when compared to conventionalelectromagnetic counterparts, has made them essential to form frequencyselective passive components in miniaturized RF platforms. Aluminumnitride (AlN) film-bulk-acoustic-resonators (FBARs) have beenextensively used to form filters in commercial RF front-ends. In fact,thanks to their ability to attain a large electromechanical couplingcoefficient (k_(t) ²˜7%), while being manufacturable throughconventional semiconductor fabrication processes, FBARs have enabledcommercial ultra- and super-high-frequency (U/SHF) acoustic filters,exhibiting fractional bandwidths and performance exceeding thoserequired by 4G communication systems.

However, despite their high-k_(t) ², the resonance frequency (f_(res))of FBARs is set by the thickness of the different forming layers. Thisfeature renders the manufacture of FBARs, having differentf_(res)-values, only attainable through post-processing steps, liketrimming and mass loading, thus leading to a much higher fabricationcomplexity. Such complexity becomes even more significant whenFBAR-based filters, operating at higher frequencies, are required tosatisfy the needs of the emerging super- and extremely-high-frequency(S/EHF) IoT and 5G applications. In fact, in such case, FBARs withsignificantly thinner metallic and piezoelectric layers are required, toenable the desired higher frequency operation.

This constraint comes with an increased sensitivity of f_(res) withrespect to the thickness of the FBAR layers. This feature renders anystep of mass-loading or trimming not easily controllable and,consequently, hardly usable in a large-scale production. Therefore, inrecent years, many groups have researched AlN-based device technologies,simultaneously enabling high-k_(t) ² and a lithographic frequencytunability. In particular, cross-sectional-Lamé-mode resonators (CLMRs)and two-dimensional-mode-resonators (2DMRs) were recently demonstrated.CLMRs excite a combination of vertical (S₁) and lateral (S₀)longitudinal motions, in AlN plates, through a coherent combination ofthe d₃₁ and d₃₃ piezoelectric coefficients. In contrast, 2DMRs excite aset of dispersive S₁-Lamb wave modes, confined between the stripsforming their metallic gratings. While CLMRs enable a comparable k_(t)²-value attained by FBARs and a significant lithographic frequencytunability (Δf), conventional 2DMRs can generally achieve a slightlylower k_(t) ² (<5%) and a reduced Δf-value. However, these devices canexcite resonant vibrations through metallic gratings that are formed bywider metallic strips than those required by CLMR, operating at the samefrequency. For this reason, they enable significantly lower ohmic lossesthan CLMRs, hence higher Q, thus being promising candidates to achievemonolithic integrated acoustic filters, for 5G communication systems.Only recently, modified 2DMRs, using a set of top and bottom metallicframes, were proposed to enable comparable k_(t) ² and Δf-valuesattained by CLMRs, while still ensuring a more relaxed lithographicresolution. However, the adoption of these frames leads to a heavilyenhanced fabrication complexity, with respect to conventional 2DMRs andCLMRs.

The operational features relative to a set of rod-modes, labeled here asdilatational modes, have been investigated through numerical methods.These modes can exhibit superior k_(t) ² (˜10%, in AlN) and Δf-values.However, up to date, it has been believed that such a high-k_(t) ² isonly attainable in one isolated and narrow rod. Consequently, the use ofdilatational modes has been considered not suitable for practicalfiltering applications, where devices with large input capacitances (C₀)are required to ensure proper functioning with 50Q-matched electronics.There has been no proposition of any multi-finger acoustic resonatordesign capable of efficiently and coherently exciting such modes, insingle piezoelectric slabs.

To address these challenges, an RF acoustic resonant technology isprovided that enables the production on integrated silicon chips oflow-loss and wideband miniaturized filters for communication systems(cellphones or other wireless devices). Devices as described herein canalso be used to attain sensors within the Internet-of-Things (IoT).Differently from previously developed technologies, devices describedherein can use the acoustic properties of profiled piezoelectric layersto achieve much better performance, in terms of electromechanicalcoupling coefficient, than attained by other devices that use plate. Forthis reason, these devices can reach performance levels that are greatlyenhanced with respect to the existing counterparts that are usednowadays in modern integrated radio-frequency front-ends.

Referring to FIGS. 1A-1C, a device 10 labeled as two-dimensionalresonant rods (“2DRRs”) is formed by a profiled AlN-layer 12, sandwichedbetween one top metal grating 14 and a grounded bottom metal plate 16(FIG. 1A). Such devices exploit the excitation of a combined set ofdilatational modes, in a single AlN plate, To do so, 2DRRs rely on aforest of locally resonant piezoelectric rods 22, built on the topsurface of a thin multi-layered beam 24. These rods are attained bypartially etching the AlN-portions to form gaps 25 between adjacentmetal strips forming the grating (FIGS. 2, 3). The etching profile ordepth is engineered to form steep trenches 26 that exhibit an evanescentlateral wavevector component (k_(x)), at f_(res). This feature permitsthe confinement of the resonant vibration within the rod, thus renderingadjacent rods only weakly and reactively coupled, so as to ensure acommon frequency of operation. Moreover, because of their modalcharacteristics, 2DRRs exhibit a higher sensitivity of f_(res) withrespect to the rod width (α). Ultimately, the adoption of trenchesallows suppression of any non-vertical electric field line that wouldotherwise be generated between adjacent strips forming the grating,which would lower the obtainable k_(t) ². Meanwhile, by adoptingun-patterned bottom metal plates instead of patterned ones, like thoserequired by both CLMRs and 2DMRs to achieve high-k_(t) ², 2DRRs can relyon AlN-films exhibiting an optimal crystalline orientation, even whenthinner films are needed to operate in the 5G spectrum. The profiled AlNlayers also render f_(res) lithographically controllable through therod-width (α, also termed W_(e)). Here, the performance of a fabricated2.3 GHz 2DRR (FIG. 1A) is reported along with discussions about itsoperation, through both a simplified one-dimensional analytical modeland through finite-element-methods (FEM).

Fabrication of the 2DRR (FIG. 1A) can follow the process flow shown inFIG. 2. In one example, it starts with the sputtering deposition of a250-nm-thick platinum full-plate 42, used as bottom electrode (FIG.2(a)), in a substrate 41. It follows with the sputtering deposition of a1-μm-thick AlN-film 44 (FIG. 2(b)). Vias 46 are then formed, in theAlN-layer, through a wet-etch process (FIG. 2(c)), followed by thesputtering and lift-off of a 330nm-thick aluminum layer 48, forming thetop metal layer (FIG. 2(d)). Then, a 150-nm-thick gold layer 52 isdeposited, on the device pads and in the vias (FIG. 2(e)), so as tominimize their associated ohmic losses. Later, the releasing holes 54are formed through an AlN dry-etch (FIG. 2(f)), followed by thepatterning of the top electrodes 56 and the derivation of the AlNtrenches 58 (FIG. 2(g)). These two steps are attained through acombination of wet and dry-etch, which can be optimized so as tominimize the surface roughness and optimize the AlN sidewall angle inthe trenches. Finally, the devices are released from the substrate 41through a XeF₂ silicon etching process (FIG. 2(h)). Tolerances on thedimensions can be ±0.5%, ±1%, ±2%, or ±5%.

In embodiments of the 2DRR described herein, a thickness of each of therod portions of the piezoelectric layer can be greater than a thicknessof the piezoelectric layer of each of the trenches. A thickness of eachof the rods can be greater than a thickness of each of the trenches.Each of the trenches can have a depth in the thickness directionselected to constrain acoustic energy within the rods. Each of the rodshas a width that can be selected to provide a determined frequency. Eachof the trenches has a width that can be selected to optimize thedetermined frequency of the rods. In some embodiments, the rods and thetrenches each have an equal width within ±1%.

To explain more particularly explain how multiple dilatational modes canbe excited in the 2DRR, the acoustic propagation characteristicsexhibited by one of its periodic cells, i.e., the unit-cell, 32 (seeFIGS. 1B, 1C) can be studied. The unit-cell is formed by two mainregions, here defined as trench 26 and rod 22. The trench is formed bythe bottom metallic plate 16 and by a thin AlN layer 12 a. The rod isformed by a thicker AlN-film 12 b of portions of the AlN layer extendingupwardly from the layer 12 a, and by the second metallic layer 14. Inthe following, the thicknesses of the thin AlN-layer 12 a, of thethicker AlN-layer 12 b, of the bottom metal layer 16 and of thetop-metal layer 14 are labeled as T_(AlN) ⁽¹⁾, T_(AlN) ⁽²⁾, T_(m) ⁽¹⁾and T_(m) ⁽²⁾, respectively. Also, the length of the unit-cell 32 islabeled as L. The interface 35 between the rod and the portion of beam24 underneath each rod is referred to as S, and Region A, 36, is theportion of the beam beneath S and the rod. The top face 34 of each rod22 behaves as a stress-free (SF) boundary. Because of the distributednature of the rod, this boundary translates into a different mechanicalboundary condition (B.C) across S, at different frequencies. Such B.Ccan significantly perturb the vertical displacement in the rod,u_(z)(z), generated by any force (F(x,z=0)) applied, to the rod, fromRegion A through S. In particular, when assuming that only a negligibledispersion affects thickness-extensional (TE) waves in the rod, F(x,z=0) can be considered uniform across S, thus being simply indicated asF. The determination of F (Eq. (8)) allows the computation of the valueof the driving impedance (Z_(b)) relative to the rod (see Eqs. (4-8)).Z_(b) allows the establishment of the influence of the rod on the B.Cexerted by Region A, across S. The distribution of Z_(b) vs. f for thefabricated 2DRR, is plotted in FIG. 6. As the trench (T_(AlN) ⁽¹⁾+T_(m)⁽¹⁾) is significantly thinner than the rod (T_(AlN) ⁽²⁾+T_(m) ⁽²⁾), itis reasonable to assume, in a first order of approximation, that onlythe flexural (A₀) and the lateral (S₀) plate modes can propagate withinthe trench. However, as the coupling between the rod and the trench canonly occur through vertical fields and since a low dispersion affectsthe velocity of the S₀-mode, for the thickness over lambda (λ) ratioused for the trench, any coupling can be neglected, through the S₀,between the rod and the trench. This simplification allows to considerthe A₀ as the only existing propagating mode that can guide acousticenergy between adjacent unit-cells of 2DRRs and permits the assumptionof a uniform transversal displacement (v(x)), throughout the thicknessof the trench. In this scenario, v(x) can be estimated by solving aone-dimensional (ID) Euler-Bernoulli equation of motion (Eq. (1)), afterselecting a proper set of B.Cs.

$\begin{matrix}{{{E_{t}I_{t}\frac{d^{4}{v(x)}}{dx^{4}}} - {\rho_{t}A_{t}\omega^{2}{v(x)}}} = {- {F\left( {{H\left( {x - \frac{a}{2}} \right)} - {H\left( {x + \frac{a}{2}} \right)}} \right)}}} & (1)\end{matrix}$

In Eq. (1), E_(t), I_(t) and A_(t) are, respectively, the effectiveYoung's modulus (Eq. 10), second moment of inertia and cross-sectionalarea relative to the trench. The function H is the Heaviside function.From the homogenous of Eq. (1), it is straightforward to estimate thereal dispersive wavevector (k, see Eq. (9)) associated to the A₀-mode,when excluding the presence of the rods. The distribution of v(x) can bederived through the methodology introduced in Cassella et al. (C.Cassella and M. Rinaldi, 2018 IEEE International Frequency ControlSymposium (IFCS) (2018), pp. 1-3). In particular, let t(x) and w(x)represent v(x), for the left and right sides of Region A (see FIG. 1C).Both displacement distributions can be expressed as the superposition ofleft/right propagating and evanescent waves (Eqs. (2-3)). It isnecessary to point out that the evanescent not propagating decayingterms can only be ignored from the solution of Eq. (1) when dealing withuniform plates, thus not including any rod.

t(x)=t _(l)(x)e ^(ikx) +t _(r)(x)e ^(ikx) +t _(re)(x)e ^(−kx) +t_(le)(x)e ^(kx)  (2)

w(x)=w _(l)(x)e ^(−ikx) +w _(r)(x)e ^(ikx) +w _(re)(x)e ^(−kx) +w_(le)(x)e ^(kx)  (3)

In Eqs. (2-3), the subscript l and r indicate the moving directions ofthe different components (from left-to-right and from right-to-left,respectively). The subscript e refers to the evanescent wave components.Investigating the wave transmission through the unit-cell requires thecomputation of a transmission-matrix, [T]_(4×4) (see Eq. (26)), relativeto the displacement field moving from one edge of the unit-cell towardsthe other. [T]_(4×4) maps the relationship between the amplitudes of thedifferent components forming Eqs. (2-3), after considering thetransformation that such components undergo, when moving through thetrench portions adjacent to Region A. From [T]_(4×4), it is easy todetermine the transmission coefficient, T, for the propagatingdisplacement component of v(x), leaving one edge of the unit-celltowards an infinite number (N→∞) of cascaded identical unit-cells. Inparticular, T is expected to be unitary at frequencies at which the roddoes not affect the propagating features of the unit-cell. In contrast,Tis expected to approach zero at the frequencies (f^((n))) at which therod exhibits the largest influence. This important feature is determinedby a process of acoustic energy storage in the rods and in Regions A.This reactive phenomenon prevents the flow of real power from adjacentunit-cells. The extrapolation of T allows to identify the existence ofpassbands and stopbands for the propagation of the A₀-mode in 2DRRs. Itis worth emphasizing that the adoption of an infinite sequence ofperiodic cells, during the evaluation of T, permits to neglect the edgeeffect that, for a finite N-value, can partially alter the validity ofthe analytical treatment. The expression of Tis rather cumbersome andits frequency distribution can only be determined numerically. As anexample, a widespan representation of T vs. f, relative to thefabricated 2DRR, is plotted in FIG. 7. As evident, multiple stopbandsexist for the A₀ propagation in the analyzed structure. In favor of aclearer visualization, the same T distribution, along with thecorresponding attenuation coefficient (R=1−|T|), is plotted in FIG.4(b), for close frequencies to the experimentally measured f_(res)(˜2.35 GHz). In order to fully understand the origin of the stopbands,both the phase (φ) and the real part (Γ_(real)) of the reflectioncoefficient (Γ) relative to the propagating displacement components, atthe right edge of the unit-cell, can be looked at. The distributions ofφ and Γ_(real) vs. f are reported in FIG. 4(a). As evident, Γ exhibits asequence of resonance conditions corresponding to the f-values at whichφ is equal to ±π. Some of them (the series resonances) correspond toΓ_(real)-values equal to −1 (i.e. the expected value for stress-freeboundaries) whereas the remaining ones (the parallel resonances)correspond to Γ_(real)-values equal to 1 (i.e. the expected value forfixed boundaries). These latter resonances identify the frequencies atwhich the rod is expected to exhibit the largest influence on thepropagation characteristics of the unit-cell. The existence of multipleun-correlated frequencies at which such strong interaction exists iscaused by the dispersive characteristics of the A₀-mode. Therefore,within the stopbands, the A₀-mode exhibits a large and evanescentwavevector (k_(ef)=N·k_(ef) ^((i)), being k_(ef) ^((i)) the wavevectorrelative to one arbitrary unit-cell) that prevents the exchange ofacoustic energy between adjacent unit-cells. In order to demonstrate theevanescent behavior of k_(ef), FIG. 4(c) reports the numerically foundreal (k_(ef-real)) and imaginary (k_(ef-im)) parts of k_(ef) ^((i)), forthe analyzed and built 2DRR. Evidently, within the stopband, k_(ef)^((i)) is purely imaginary, which is a direct proof that no propagationof the A₀ occurs within this frequency range.

Despite the fact that one side of an arbitrary unit-cell was used as thereference location (x₀) for the computation of T, the magnitude of T isinvariant to x₀. So, the same T-values would be attained if a differentreference location, included in the Region A, were used. Thisconsideration helps to understand the origin of the unique modalfeatures that characterize the operation of 2DRRs (see the FEM simulateddisplacement modeshape in FIG. 5A. In fact, the resonant vibration ofthese devices is piezoelectrically generated, in the rods and in RegionsA, from the vertical electric field (E_(z)) that exists between the topmetal strip and the bottom metal plate. In particular, E_(z) couples tomechanical strain through the AlN d₃₁ and d₃₃ piezoelectriccoefficients. However, because of the described dispersive properties ofthe unit-cell, the lateral edges of Regions A, from which the acousticenergy would tend to leak towards adjacent unit-cells, behave asfixed-boundaries. Consequently, the generated acoustic energy comes tobe stored in the rod structures, whose lateral sides act as SFboundaries, hence being more prone to deform. As experimentallydemonstrated (discussed below), this operational feature allows thegeneration of more mechanical energy than possible when no trench isused, thus being ultimately the main feature responsible for thehigh-k_(t) ² attained by 2DRRs.

The electrical performance of the fabricated 2DRR (details on thefabrication flow described above) were extracted through conventional RFcharacterization tools. This device, which is formed by 20 unit-cells,shows measured k_(t) ², resistance at resonance (R_(tot)=R_(m)+R_(s)),loaded quality factor Q_(3dB) (extracted from the 3 dB bandwidth) and C₀in excess of 7.4%, 56 Ω, 185 and 325 fF (corresponding to an impedanceof 208 Ω), respectively. The measured k_(t) ² and C₀ values matchclosely their FEM predicted values (7.7% and 300 fF, respectively).Ultimately, the capability to lithographically define the resonancefrequency of 2DRRs was also investigated through FEM (FIG. 5C). This wasdone by simulating the trends of k_(t) ² and f_(res) vs. α, whenconsidering the same material stack adopted for the reported 2DRRdevice. As evident, 2DRRs simultaneously enable a significantlithographic frequency tunability (Δf>117 MHz) and a large k_(t) ²exceeding 5%. This feature renders them suitable components to formmonolithic integrated wideband filters, for next-generation RFfront-ends.

The devices can provide several advantages. Their operation relies on anacoustic technology that has not been exploited. The devices can exciteacoustic waves in profiled aluminum nitride layers rather than inplates. This feature allows to overcome the technological limits thatcurrently characterize the performance of existing radio-frequencyaluminum nitride devices. This same technology can be applied to manyother piezoelectric materials, such as PZT, LiNbO₃, aluminum thiocyanate(AlScN), and others. The devices not only allow the achievement ofelectromechanical coupling coefficients that can be higher than thoseattained by the commercialized ones, but also permit the building ofmultiple resonators on the same chip, thus creating an opportunity tobuild monolithic integrated filters, in commercial radio-frequencyfront-ends. The acoustic characteristics and operation of these devicesshow significant differences with respect to the other previouslydeveloped acoustic resonant technologies.

The devices described herein can achieve a higher electromechanicalcoupling coefficient (k_(t) ²) exceeding what is currently possiblethrough the other available technologies. This feature permits thebuilding of filters with wider fractional bandwidth and enablesoscillators capable to dissipate a lower amount of energy to operate.The devices are lithographically defined aluminum nitride (AlN) acoustictechnologies that have the ability to achieve high k_(t) ² (>7%) whilenot requiring any patterned bottom-metal structure underneath theAlN-layer. For this reason, they do not suffer from any performancedegradation that occurs, instead, in previously developedlithographically defined technologies and that are caused by the need ofdepositing the AlN-layer on top of a patterned metal structure. Thisfeature solves a challenge that has prevented the use oflithographically defined AlN-acoustic resonators in the past. Thedevices use lithographic resolutions that can be more relaxed thanrequired by previously developed AlN lithographically definedtechnologies. This feature enables their fabrication through standardlithographic methods, thus opening them to large-scale production.

The devices described herein have performance that make them useful tobuild low-loss and wideband RF filters for integrated wirelessfront-ends. Thus, they can be used to replace the currently usedacoustic resonator technologies from next-generation radio-frequencyfront-ends. The devices have usefulness in the generation of low-powerhigh-frequency oscillators for timing applications. Due to the superiorelectromechanical coupling (k_(t) ²), the devices can lead to areduction in the amount of active gain that is required by activecomponents in oscillators to sustain a low-noise oscillation. Thetechnologies can have application as new sensors directly relying on thetopology of the structure to reach higher sensitivity and lower powerconsumption. Because of their high coupling coefficient, the devicesallow to build other RF passive components, such as hybrid couplers,piezoelectric transformers and others.

The devices can be used in applications related to the generation oflow-power RF oscillators for timing applications. Oscillators are usedby wireless systems to enable the required frequency conversion stagesthat permit to code or decode the information that must be transmittedor received, respectively. In addition, recent efforts have led to thefirst AlN-based voltage-controlled-oscillators in RF frequencysynthesizers. These synthesizers are based on the use offilm-bulk-acoustic-resonators, thus exhibiting a lower tuningsensitivity than possible through the adoption, in the same systems, ofthe devices described herein.

By enabling the fabrication of multiple resonators having differentfrequencies, through the same fabrication process, the design andfabrication complexity of needed RF filters can be simplified. Thisfeature can lead to less expensive costs of production for suchcomponents.

The high electromechanical coupling coefficient that these deviceexhibit, together with their lithographic frequency tunability, renderthese devices useful as a passive technology for next-generationradio-frequency front-ends.

Analytical Study of the 2DRR Unit-Cell

In order to analyze the operation and unconventional dispersivecharacteristics of 2DRRs, it suffices to investigate the acousticbehavior relative to one of their periodic cell (i.e., the unit-cell,see FIGS. 1B, 1C). In fact, such cell captures all transitions betweendifferent acoustic characteristics that periodically occur across theentire device geometry. As discussed above, the unit-cell is formed bytwo main regions, defined as trench and rod, which are characterized,for the 2DRR fabricated according to this technology, by the mechanicaland geometrical parameters shown in FIGS. 1B, 1C.

The faces of each rod that are orthogonal to their main vibrationaldirection (vertical z-direction) are characterized by different boundaryconditions (B.Cs). The top face behaves as a stress-free boundary,whereas the bottom face is directly attached to the trench regionlabeled as Region A (see FIG. 1C). In particular, this latter face isloaded by the longitudinal force (F see Eq. (1) above) originated inRegion A, oriented along the z-direction and perturbing the stressdistribution and propagation features relative to the entire unit-cell.When assuming that only a negligible dispersion affects the velocity oflongitudinal thickness-extensional (TE) waves in the rod, u_(z)(z) canbe estimated, in its frequency-domain representation, as (Eq. (4)):

$\begin{matrix}{{u_{z}(z)} = {{- \frac{F}{\omega \; \rho_{rod}A_{rod}c_{rod}}}\left( {{\sin \left( {k_{rod}z} \right)} + \frac{\cos \left( {k_{rod}z} \right)}{\tan \left( {k_{rod}\left( {T_{AlN}^{(2)} + T_{m}^{(2)}} \right)} \right)}} \right)}} & (4)\end{matrix}$

where F can be rewritten as −E_(rod)A_(rod)ϵ_(z) (z=0), being ϵ_(z)(z=0) the strain across S. In Eq. (1) ρ_(rod), A_(rod) and c_(rod)represent the effective mass density, the cross-sectional area (i.e.(T_(AlN) ⁽²⁾+T_(m) ⁽²⁾)·W, being W the out-of-plane dimension relativeto both the trenches and the rods) and the nondispersive phase velocity,for longitudinal waves, in the rods. In contrast, k_(rod) and ωrepresent the wavevector relative to the same vertical motion and thenatural frequency (i.e. ω=3πf, being f the frequency), respectively. Inorder to estimate the mechanical properties and resonance frequency(f_(res)) of the TE-mode, in the rod, ρ_(rod) and c_(rod) can be foundafter computing effective values for the Young's modulus (E_(rod)) andmass density, based on the geometrical and mechanical parametersrelative to the materials forming the rods. Therefore, E_(rod) andρ_(rod) can be estimated as (Eqs. (5-8)):

$\begin{matrix}{E_{rod} = \frac{\left( {{E_{AlN}T_{AlN}^{(2)}} + {E_{m}^{(2)}T_{m}^{(2)}}} \right)}{T_{m}^{(2)} + T_{AlN}^{(2)}}} & (5) \\{\rho_{rod} = \frac{\left( {{\rho_{AlN}T_{AlN}^{(2)}} + {\rho_{m}^{(2)}T_{m}^{(2)}}} \right)}{T_{m}^{(2)} + T_{AlN}^{(2)}}} & (6)\end{matrix}$

From Eqs. (2-6), c_(rod) can be found as:

$\begin{matrix}{c_{rod} = \sqrt{\frac{E_{rod}}{\rho_{rod}}}} & (7)\end{matrix}$

The driving impedance across S (Z_(b)), relative to each rod, can befound through the Mason formalism (Eq. (8).

$\begin{matrix}{Z_{b} = {\frac{F}{{vel}\left( {z = 0} \right)} = {\frac{F}{{- i}\omega {u_{z}\left( {z = 0} \right)}} = {{- i}\; \rho_{rod}A_{rod}c_{rod}{\tan \left( {k_{rod}\left( {T_{AlN}^{(2)} + T_{m}^{(2)}} \right)} \right)}}}}} & (8)\end{matrix}$

In Eq. (8), vel(z=0) is the magnitude of the laterally uniform verticalvelocity (i.e. time derivative of u_(z)(z=0) with respect to time), atS. It is straightforward to notice (Eq. (6)) that Z_(b) exhibits both alocal maximum and a local minimum, at two correlated frequencies,f_(min) and f_(max), respectively. In particular, for f equal tof_(min), Z_(b) is equal to zero. Thus, at this frequency of operation,the rod does not exert any constraint on the displacement at S. For thisreason, S acts, at f_(min), as a conventional SF boundary, placed in theactive resonator portion. In contrast, for f equal to f_(max), the rodimposes a virtual fixed-constraint across S. It is worth mentioning thatother non-conventional B.Cs characterize the impact of the rod on thebehavior of S and, consequently, of Region A, for different frequenciesfrom f_(min) and f_(max). Eq. (8) reports the distribution of Z_(b), forthe material and geometrical characteristics reported in FIG. 1discussed above.

As discussed above, the propagation characteristics within the unit-cellcan be found by solving the 1D Euler-Bernoulli equation (see Eq. (1)above), in terms of the transversal displacement in the trench (v(x)),after properly selecting a suitable set of B.Cs. However, from thesolution of the homogenous of Eq. (1), it is useful to extract thewavevector (Eq. (9)) associated to the A₀-mode, when neglecting thepresence of the rods.

$\begin{matrix}{k = \frac{\sqrt{2}3^{1/4}\sqrt{\omega}\rho_{t}^{1/4}}{\sqrt{T_{m}^{(1)} + T_{AlN}^{(1)}}E_{t}^{1/4}}} & (9)\end{matrix}$

In Eq. (9), E_(t) and ρ_(t) are effective Young's modulus and massdensity relative to the trench. In analogy to that done for the rods(Eqs. (5-6)), these parameters can be found, for the unit-cell of thefabricated 2DRR (FIGS. 1B, 1C), as (Eqs. (10-11)):

$\begin{matrix}{E_{t} = \frac{\left( {{E_{AlN}T_{AlN}^{(1)}} + {E_{Pt}T_{m}^{(1)}}} \right)}{T_{m}^{(1)} + T_{AlN}^{(1)}}} & (10) \\{\rho_{t} = \frac{\left( {{\rho_{AlN}T_{AlN}^{(1)}} + {\rho_{Pt}T_{m}^{(1)}}} \right)}{T_{m}^{(1)} + T_{AlN}^{(1)}}} & (11)\end{matrix}$

It can be pointed out that the magnitude of F, in Eq. (1) above, can bedirectly expressed in terms of the driving impedance of the rod (Z_(b),see Eq. (8)). In fact, its value can be computed as F=Z_(b)·vel(z=0),thus being independent from x. The distribution of v(x) can be derivedthrough the methodology discussed in Williams et al. 2015 (E. G.Williams, P. Roux, M. Rupin, and W. A. Kuperman, Phys. Rev. B 91, 104307(2015)). In particular, after breaking v(x) into its portions (t(x) andw(x)), for the left and right sides of Region A (see Eqs. (2-3) above),a scattering matrix ([G]_(4×4)) can be defined. [G]_(4×4) captures thechanges of the wave characteristics (Eq. (12)) that occur at thetransitions between each uncovered trench region and Region A (see FIGS.1B, 1C). More specifically, [G]_(4×4) allows to map the interactionbetween the wave components going towards the rods (t_(l), t_(le),w_(r), w_(re)) and those (t_(r), t_(re), w_(l) and w_(le)) that,instead, are reflected by them r, re, (Eq. (12)). For this reason, it isa function of the geometrical and material composition of the entireunit-cell. The matrix [G]_(4×4) is reported in Eq. (13).

$\begin{matrix}{\begin{bmatrix}t_{l} \\t_{le} \\w_{r} \\w_{re}\end{bmatrix} = {\lbrack G\rbrack_{4 \times 4}\begin{bmatrix}t_{r} \\t_{re} \\w_{l} \\w_{le}\end{bmatrix}}} & (12) \\{\lbrack G\rbrack_{4 \times 4} = \begin{bmatrix}r & r_{ef} & t & t_{ef} \\r_{fe} & r_{e} & t_{fe} & t_{e} \\t & t_{ef} & r & r_{ef} \\t_{fe} & t_{e} & r_{fe} & r_{e}\end{bmatrix}} & (13)\end{matrix}$

It can be pointed out that the two vectors shown in theleft-([v₁]_(4×4)) and right-([v₂]_(4×4)) sides of Eq. (12) are composedby the amplitudes of the different wave components forming t(x) andw(x), at the lateral edges of Region A. Regarding [G]_(4×4), r and t mapthe reflection and transmission coefficients for the differentpropagating wave components that are incident towards Region A. Clearly,due to the symmetric nature of this problem, r and t have the same valuefor both t(x) and w(x). Similarly, r_(e) and t_(c) represent thereflection and transmission coefficients relative to the evanescentterms of t(x) and w(x). Ultimately, r_(ef), t_(ef), r_(fe) and t_(fe)represent reflection and transmission coefficients capturing thephenomenon of energy-exchange between wave components having differentpropagation characteristics. Such wave-conversion phenomenon isoriginated from the different dispersive characteristics relative todistinct unit-cell regions (FIGS. 1B, 1C). In particular, r_(ef) andt_(ef) map the amplitude change relative to the reflected/transmittedflexural wave components, originated from the evanescent ones that areincident towards Region A. Similarly, r_(fe) and t_(fe) capture theamplitude change relative to the reflected/transmitted evanescent wavecomponents, originated from the propagating ones that are incidenttowards Region A. The described transmission and reflection coefficientscan be found by applying suitable boundary conditions (B.Cs) to Eq. (1).In particular, when assuming that the coupling of the rod with Region Acan only occur through longitudinal vertical mechanical fields, the rodcan only displace like a piston (pure TE). As a result, a uniformlateral displacement profile is expected in Region A. In such scenario,the B.Cs shown in Eq. (14) can be used to approximate the expected modalcharacteristics under the rod. In particular, the first three equationsof Eq. (14) map the equality in the displacement, slope and curvature atthe edges of Region A.

$\begin{matrix}{{t\left( {- \frac{a}{2}} \right)} = {w\left( \frac{a}{2} \right)}} & (14) \\{t^{\prime {({- \frac{a}{2}})}} = {w^{\prime}\left( \frac{a}{2} \right)}} & \; \\{t^{''{({- \frac{a}{2}})}} = {w^{''}\left( \frac{a}{2} \right)}} & \; \\{{w^{{\prime\prime\prime}{({- \frac{a}{2}})}} - {t^{\prime\prime\prime}\left( \frac{a}{2} \right)}} = {\frac{F}{E_{t}I_{t}} = \frac{Z_{b}{{vel}\left( {z = 0} \right)}}{E_{t}I_{t}}}} & \;\end{matrix}$

In contrast, the fourth equation captures the existence of ashear-force, at S, that counterbalances the laterally uniform forcedistribution at the bottom surface of the rod. After substituting Eqs.(2-3) (see above) in Eq. (14), the reflection and transmissioncoefficients, discussed above, can be found (see Eqs. (15-20)).

$\begin{matrix}{r = {- \frac{\left( {1 - i} \right)e^{{- i}ak}F}{{2F} + {\left( {4 + {4i}} \right)E_{t}l_{t}k^{3}}}}} & (15) \\{t = \frac{\left( {\frac{1}{2} + \frac{i}{2}} \right){e^{{- i}ak}\left( {F + {4E_{t}I_{t}k^{3}}} \right)}}{F + {\left( {2 + {2i}} \right)E_{t}I_{t}k^{3}}}} & (16) \\{r_{ef} = {t_{ef} = {- \frac{\left( {1 - i} \right)e^{{({\frac{1}{2} - \frac{i}{2}})}{ak}}F}{{2F} + {\left( {4 + {4i}} \right)E_{t}I_{t}k^{3}}}}}} & (17) \\{r_{fe} = {t_{fe} = {- \frac{\left( {1 + i} \right)e^{{({\frac{1}{2} - \frac{i}{2}})}{ak}}F}{{2F} + {\left( {4 + {4i}} \right)E_{t}I_{t}k^{3}}}}}} & (18) \\{r_{e} = {- \frac{\left( {1 + i} \right)e^{ak}F}{{2F} + {\left( {4 + {4i}} \right)E_{t}I_{t}k^{3}}}}} & (19) \\{t_{e} = \frac{\left( {\frac{1}{2} - \frac{i}{2}} \right){e^{ak}\left( {F + {4iE_{t}I_{t}k^{3}}} \right)}}{F + {\left( {2 + {2i}} \right)E_{t}I_{t}k^{3}}}} & (20)\end{matrix}$

It is worth pointing out that, when F is equal to zero, thus indicatingthat the rod does not affect the propagation of flexural waves in theunit-cell, [G]_(4×4) becomes (Eq. (21)):

$\begin{matrix}{\lbrack G\rbrack_{4 \times 4} = \begin{bmatrix}0 & 0 & e^{{- i}ka} & 0 \\0 & 0 & 0 & e^{ka} \\e^{{- i}ka} & 0 & 0 & 0 \\0 & e^{ka} & 0 & 0\end{bmatrix}} & (21)\end{matrix}$

Therefore, as expected, in such simplified scenario, all the reflectioncoefficients, as well as the transmission coefficients associated to theprocess of wave-conversion (i.e. t_(ef) and t_(fe)), become zero.Therefore, only the transmission coefficient (t) is not nulled and equalto its expected value after assuming Region A to only act as an acousticdelay line that phase shifts or attenuates any existing propagating andevanescent component of t(x) and w(x), by an amount that is proportionalto α. It is now useful to manipulate [G]_(4×4) in such a way that theamplitudes of the wave components coming from the left-side of the rod(i.e. [t]_(4×1)=[t_(l), t_(le), t_(r), t_(re)]^(T)) become theindependent variables of Eq. (12), whereas those outgoing the right-sideof the rod (i.e. [w]_(4×1)=[w_(l), w_(le), w_(r), w_(re)]^(T)) act asthe dependent ones. In such scenario, Eq. (12) becomes:

$\begin{matrix}{\begin{bmatrix}w_{l} \\w_{le} \\w_{r} \\w_{re}\end{bmatrix} = {\lbrack C\rbrack_{4 \times 4}\begin{bmatrix}t_{l} \\t_{le} \\t_{r} \\t_{re}\end{bmatrix}}} & (22)\end{matrix}$

In Eq. (22), the matrix [C]_(4×4), known as the coupling matrix,captures the wave transmission characteristics (from the left-side tothe right-side of the rod). The derived expression for [C]_(4×4) isreported in Eq. (23).

$\begin{matrix}{\lbrack C\rbrack_{4 \times 4} = \begin{bmatrix}{- \frac{{ie}^{iak}\left( {F + {4{ik}^{3}}} \right)}{4\; k^{3}}} & {- \frac{{ie}^{{({{- \frac{1}{2}} + \frac{i}{2}})}{ak}}F}{4\; k^{3}}} & {- \frac{iF}{4k^{3}}} & {- \frac{{ie}^{{({\frac{1}{2} + \frac{i}{2}})}{ak}}F}{4\; k^{3}}} \\\frac{e^{{({{- \frac{1}{2}} + \frac{i}{2}})}{ak}}F}{4\; k^{3}} & \frac{e^{- {ak}}\left( {F + {4k^{3}}} \right)}{4\; k^{3}} & \frac{e^{{({{- \frac{1}{2}} - \frac{i}{2}})}{ak}}F}{4\; k^{3}} & \frac{F}{4k^{3}} \\\frac{iF}{4k^{3}} & \frac{{ie}^{{({{- \frac{1}{2}} - \frac{i}{2}})}{ak}}F}{4\; k^{3}} & \frac{{ie}^{- {iak}}\left( {F - {4{ik}^{3}}} \right)}{4\; k^{3}} & \frac{{ie}^{{({{- \frac{1}{2}} - \frac{i}{2}})}{ak}}F}{4\; k^{3}} \\{- \frac{e^{{({\frac{1}{2} + \frac{i}{2}})}{ak}}F}{4\; k^{3}}} & {- \frac{F}{4k^{3}}} & {- \frac{e^{{({\frac{1}{2} - \frac{i}{2}})}{ak}}F}{4\; k^{3}}} & {- \frac{e^{ak}\left( {F - {4k^{3}}} \right)}{4\; k^{3}}}\end{bmatrix}} & (23)\end{matrix}$

The derivation of [C]_(4×4) is key to find the transmission matrix,[T]_(4×4), relative to the entire unit-cell. However, in order to do so,it is necessary to apply an additional boundary condition that forces aperiodic displacement distribution, with period equal to L, betweenadjacent periodic cells. This can be done by defining two newdisplacement vectors, [t⁻]_(4×1) and [w₊]_(4×1), for t(x=−L/2) andw(x=L/2). In particular, when neglecting the existence of any lossmechanism, [t⁻]_(4×1) and [w₊]_(4×1) are equal in magnitude andequivalent to modified versions of [t]_(4×4) and [w]_(4×4). Suchmodified versions are formed by phase-shifted or more attenuated copiesof the propagating and evanescent components forming [t]_(4×1) and[w]_(4×1), respectively (see Eqs. (24-25)):

$\begin{matrix}{\begin{bmatrix}w_{l, +} \\w_{{le}, +} \\w_{r, +} \\w_{{re}, +}\end{bmatrix} = {{\lbrack D\rbrack_{4 \times 4}\begin{bmatrix}w_{l} \\w_{le} \\w_{r} \\w_{re}\end{bmatrix}} = {\begin{bmatrix}e^{{- i}\varphi} & 0 & 0 & 0 \\0 & e^{\varphi} & 0 & 0 \\0 & 0 & e^{i\varphi} & 0 \\0 & 0 & 0 & e^{- \varphi}\end{bmatrix}\begin{bmatrix}w_{l} \\w_{le} \\w_{r} \\w_{re}\end{bmatrix}}}} & (24) \\{\begin{bmatrix}t_{l, -} \\t_{{le}, -} \\t_{r, -} \\t_{{re}, -}\end{bmatrix} = {{\lbrack D\rbrack_{4 \times 4}^{- 1}\begin{bmatrix}t_{l} \\t_{le} \\t_{r} \\t_{re}\end{bmatrix}} = {\begin{bmatrix}e^{{- i}\varphi} & 0 & 0 & 0 \\0 & e^{\varphi} & 0 & 0 \\0 & 0 & e^{i\varphi} & 0 \\0 & 0 & 0 & e^{- \varphi}\end{bmatrix}^{- 1}\begin{bmatrix}t_{l} \\t_{le} \\t_{r} \\t_{re}\end{bmatrix}}}} & (25)\end{matrix}$

In Eqs. (24-25), φ is equivalent to k·(L+α)/2. From Eqs. (24-25) it ispossible to compute [T]_(4×4), relative to all the wave componentstravelling between adjacent edges of the unit-cell. This can be done byusing Eq. (26).

$\begin{matrix}{\lbrack T\rbrack_{4 \times 4} = {{\lbrack D\rbrack_{4 \times 4} \cdot \lbrack C\rbrack_{4 \times 4} \cdot \lbrack D\rbrack_{4 \times 4}}==\begin{bmatrix}\frac{e^{- {ikL}}\left( {{- {iF}} + {4k^{3}}} \right)}{4\; k^{3}} & {- \frac{{ie}^{{({\frac{1}{2} - \frac{i}{2}})}{kL}}F}{4\; k^{3}}} & {- \frac{iF}{4k^{3}}} & {- \frac{{ie}^{{({{- \frac{1}{2}} - \frac{i}{2}})}{kL}}F}{4\; k^{3}}} \\\frac{e^{{({\frac{1}{2} - \frac{i}{2}})}{kL}}F}{4\; k^{3}} & {\frac{1}{4}{e^{kL}\left( {4 + \frac{F}{k^{3}}} \right)}} & \frac{e^{{({\frac{1}{2} + \frac{i}{2}})}{kL}}F}{4\; k^{3}} & \frac{F}{4k^{3}} \\\frac{iF}{4k^{3}} & \frac{{ie}^{{({\frac{1}{2} + \frac{i}{2}})}{kL}}F}{4\; k^{3}} & \frac{e^{ikL}\left( {{iF} + {4k^{3}}} \right)}{4\; k^{3}} & \frac{{ie}^{{- {ak}} - {{({\frac{1}{2} - \frac{i}{2}})}{kL}}}F}{4\; k^{3}} \\{- \frac{e^{{({{- \frac{1}{2}} - \frac{i}{2}})}{kL}}F}{4\; k^{3}}} & {- \frac{F}{4k^{3}}} & {- \frac{e^{{({{- \frac{1}{2}} + \frac{i}{2}})}{kL}}F}{4\; k^{3}}} & {\frac{1}{4}{e^{- {kL}}\left( {4 - \frac{F}{k^{3}}} \right)}}\end{bmatrix}}} & (26)\end{matrix}$

As a sanity check, it is useful to look at the value of [T]_(4×4) when Fis set to be zero, thus when the rod does not perturb the propagationcharacteristics of the unit-cell. As evident, in such scenario, theexpression of [T]_(4×4) is heavily simplified (see Eq. (27)), thusclearly mapping the case in which the unit-cell can only phase-shift orattenuate existing propagating and evanescent wave components, in thetrench.

$\begin{matrix}{\lbrack T\rbrack_{4 \times 4} = \begin{bmatrix}e^{{- i}kL} & 0 & 0 & 0 \\0 & e^{kL} & 0 & 0 \\0 & 0 & e^{ikL} & 0 \\0 & 0 & 0 & e^{{- k}L}\end{bmatrix}} & (27)\end{matrix}$

It is important to point out that, as expected, different frequencybehaviors characterize corresponding components of [T]_(4×4) and[C]_(4×4). Such a unique feature, which is mostly determined by thenonlinear dependence of k with respect to frequency (see Eq. (9)),determines the existence of multiple not correlated frequencies(f^((n))) at which the rod exerts the largest influence on thepropagation capability of the trench. In particular, from the analysisof the eigenvalues of [T]_(4×4) (see Eq. (26)), it is possible todetermine the transmission coefficient, T.

As discussed above, this coefficient captures the reduction of themagnitude of a propagating displacement component, referring to one edgeof an arbitrary chosen unit-cell and outgoing the same cell towards aninfinite number (N→∞) of cascaded and identical unit-cells. As anexample, T is plotted, in FIG. 8, when assuming the same unit-cellgeometry and material stack adopted in the fabricated 2DRR. As evident,multiple stopbands exist for the propagation of the A₀ in the analyzedstructure. It is straightforward to notice that the center frequenciesof such forbidden bands closely match the f^((n)) (from 1 to 8) values,identified in FIG. 4(a), above. This clearly proves that the largestinfluence of the rods, on the propagation characteristics of theunit-cell, occurs at those frequencies at which each rod producesvirtual fixed-constraints across the lateral edges of the correspondingunit-cell.

In order to clearly visualize the evanescent behavior of theinvestigated unit-cell, an ad-hoc FEM simulation framework was createdto analyze the propagation characteristics exhibited by a chain ofunit-cells, for frequencies included in three expected stopbands. Thisframework uses different piezoelectric generators to producelongitudinal vibrations at significantly different frequencies (2.4 GHz,1.7 GHz and 500 MHz). This generator is attached, at one of its lateralside, to a perfectly-matched-layer (PML) while being connected, on theopposite side, to a chain of seven additional and electrically floatingunit-cells (FIGS. 8-10). This chain acts as a delay line, separating thegenerator from an additional PML. FIGS. 8-10 report the generatedmodeshape relative to the magnitude of the total-displacement across thechain of unit-cells, for the three investigated frequencies. Also, thedistribution of the total displacement along a cut horizontal line,starting from the top-left edge of the first unit-cell of the chain andending at the edge of the furthest PML, is also reported, for the sameinvestigated cases.

As evident, the simulated displacement profiles, across the unit-cellsand relative to the three investigated cases, clearly exhibit thetypical exponential decay that is expected by a delay-line operating inits evanescent operational region.

Measured Impact of the Trenches on k_(t) ²

To verify the high-k_(t) ² attained by 2DRRs, a low-impedance((ωC₀)⁻¹≈208 Ω) 2.35 GHz 2DRR (Device A) exhibiting a k_(t) ²-value inpair of 7.4% was built (FIGS. 3, 13). This device, which is formed by 20unit-cells, shows measured k_(t) ², resistance at resonance(R_(tot)=R_(m)+R_(s)), loaded quality factor Q_(3dB) (extracted from the3 dB bandwidth) and C₀ in excess of 7.4%, 56 Ω, 185 and 325 fF(corresponding to an impedance of 208 Q), respectively. Such a highk_(t) ²-value matches closely the predicted one (7.2%) found throughFinite Element Methods (FEM). In addition, the capability tolithographically define the resonance frequency of 2DRRs, without asubstantial degradation of k_(t) ² and without requiring additionalfabrication steps, was demonstrated through FEM (FIG. 15).

In order to experimentally demonstrate the impact of the trenches on theattainable k_(t) ², a second device (Device B) (FIG. 14; see also FIG.11), with the same geometrical and material characteristics, but notrelying on any trench, was simultaneously fabricated on the same siliconwafer than the reported 2DRR. As expected, this device showed asignificantly lower k_(t) ² (<4.4%) than attained by the fabricated2DRR. Also, it showed a resistance at resonance (R_(tot)=R_(m)+R_(s),being R_(m) and R_(s) its motional and series resistance, respectively),a loaded quality factor Q_(3dB) (extracted from the 3 dB bandwidth) anda C₀-value (being its static capacitance) in excess of 64 Q, 236 and 330fF, respectively. The measured k_(t) ² and C₀ values for this devicematch closely their FEM predicted values (4.6%, 340 fF respectively).Also, the measured Q_(3dB)-value for this device matches well the oneattained by the reported 2DRR (see FIG. 3 described above). This factclearly shows that the quality factor that was found for both thisdevice and the 2DRR reported above is not limited by the presence of thetrenches.

As used herein, “consisting essentially of” allows the inclusion ofmaterials or steps that do not materially affect the basic and novelcharacteristics of the claim. Any recitation herein of the term“comprising,” particularly in a description of components of acomposition or in a description of elements of a device, can beexchanged with “consisting essentially of” or “consisting of”

To the extent that the appended claims have been drafted withoutmultiple dependencies, this has been done only to accommodate formalrequirements in jurisdictions that do not allow such multipledependencies. It should be noted that all possible combinations offeatures that would be implied by rendering the claims multiplydependent are explicitly envisaged and should be considered part of theinvention.

The present technology has been described in conjunction with certainpreferred embodiments and aspects. It is to be understood that thetechnology is not limited to the exact details of construction,operation, exact materials or embodiments or aspects shown anddescribed, and that various modifications, substitution of equivalents,alterations to the compositions, and other changes to the embodimentsand aspects disclosed herein will be apparent to one of skill in theart.

What is claimed is:
 1. A resonator device comprising: a piezoelectriclayer suspended from a substrate, the piezoelectric layer having a widthdirection, a length direction, and a thickness direction, thepiezoelectric layer having a bottom surface and a top surface, thepiezoelectric layer comprising a beam extending along the bottom surfacecontinuously in the width direction and the length direction, aplurality of rod portions extending continuously in the length directionand upwardly from the beam to the top surface, the rod portions spacedapart in the width direction; a bottom electrode comprising a metalplate in contact with the bottom surface of the piezoelectric layer; anda top electrode comprising a metal grating comprising a plurality ofconductive strips on the top surface of the piezoelectric layer, each ofthe conductive strips aligned in contact with a corresponding one of therod portions of the piezoelectric layer; wherein the plurality ofconductive strips and corresponding ones of the rod portions of thepiezoelectric layer form a plurality of rods, and portions of the beamof the piezoelectric layer and the bottom electrode form a plurality oftrenches between the rods.
 2. The resonator device of claim 1, wherein athickness of each of the rods is greater than a thickness of each of thetrenches.
 3. The resonator device of claim 1, wherein a thickness ofeach of the rod portions of the piezoelectric layer is greater than athickness of the piezoelectric layer of each of the trenches.
 4. Theresonator device of claim 1, wherein each of the trenches has a depth inthe thickness direction selected to constrain acoustic energy within therods.
 5. The resonator device of claim 1, wherein each of the rods has awidth selected to provide a determined frequency.
 6. The resonatordevice of claim 5, wherein each of the trenches has a width selected tooptimize the determined frequency of the rods.
 7. The resonator deviceof claim 1, wherein the rods and the trenches each have an equal widthwithin ±1%.
 8. The device of claim 1, further comprising circuitry incommunication with the resonator device to apply an alternating voltagethrough the top electrode and the bottom electrode to excite one or moredilatational modes of mechanical vibration in the rods of thepiezoelectric layer.
 9. The resonator device of claim 1, wherein thepiezoelectric layer is selected from aluminum nitride, lead zirconatetitanate, lithium niobate, lithium tantalite, zinc oxide, galliumnitride, scandium nitride, aluminum scandium nitride, and quartz, andcombinations thereof.
 10. The resonator device of claim 1, wherein thetop electrode and the bottom electrode are each selected from aluminum,platinum, ruthenium, molybdenum, tungsten, and gold, and combinationsthereof.
 11. The resonator device of claim 1, wherein the piezoelectriclayer is aluminum nitride, the bottom electrode is platinum, and the topelectrode is aluminum.
 12. The resonator device of claim 1, wherein thedevice has an electromechanical coupling coefficient between electricaland acoustic energy of at least 7%.
 13. A filter comprising theresonator device of claim 1, and electrical connections to the resonatordevice.
 14. A radio frequency (RF) component including the resonatordevice of claim 1, and electrical connections to the resonator device,wherein the RF component is a filter, oscillator, synthesizer, sensor,coupler, or transformer.
 15. A method of fabricating a resonator device,comprising: forming a bottom metal layer on a substrate to form a bottomelectrode; forming a piezoelectric material on the bottom metal layer;forming a top metal layer on the piezoelectric material comprising agrating of conductive strips to form a top electrode; removing materialfrom the piezoelectric material between the conductive strips of the topelectrode to form a plurality of alternating rods and trenches; andreleasing the bottom electrode, the piezoelectric material and the topelectrode from the substrate.
 16. The method of claim 15, wherein thebottom electrode is formed from aluminum, platinum, ruthenium,molybdenum, tungsten, or gold, or combinations thereof.
 17. The methodof claim 15, wherein the top electrode is formed from aluminum,platinum, ruthenium, molybdenum, tungsten, or gold, or combinationsthereof.
 18. The method of claim 15, wherein the piezoelectric materialis formed from aluminum nitride, lead zirconate titanate, lithiumniobate, lithium tantalite, zinc oxide, gallium nitride, scandiumnitride, aluminum scandium nitride, or quartz, or combinations thereof.19. The method of claim 15, further comprising forming one or more viasin the piezoelectric layer to the bottom electrode, and depositing ametal material in the one or more vias and on device pads.